Contrasting the Gini and Zenga indices of economic inequality

The current financial turbulence in Europe inspires and perhaps requires researchers to rethink how to measure incomes, wealth, and other parameters of interest to policy-makers and others. The noticeable increase in disparities between less and more fortunate individuals suggests that measures based upon comparing the incomes of less fortunate with the mean of the entire population may not be adequate. The classical Gini and related indices of economic inequality, however, are based exactly on such comparisons. It is because of this reason that in this paper we explore and contrast the classical Gini index with a new Zenga index, the latter being based on comparisons of the means of less and more fortunate sub-populations, irrespectively of the threshold that might be used to delineate the two sub-populations. The empirical part of the paper is based on the 2001 wave of the European Community Household Panel data set provided by EuroStat. Even though sample sizes appear to be large, we supplement the estimated Gini and Zenga indices with measures of variability in the form of normal, t-bootstrap, and bootstrap bias-corrected and accelerated confidence intervals.     

Zenga Distribution and Inequality Ordering

The aim of this article is to establish an ordering related to the inequality for the recently introduced Zenga distribution. In addition to the well-known order based on the Lorenz curve, the order based on I(p) curve is considered. Since the Zenga distribution seems to be suitable to model wealth, financial, actuarial, and, especially, income distributions, these findings are fundamental in the understanding of how parameter values are related to inequality. This investigation shows that for the Zenga distribution, two of the three parameters are inequality indicators.     

On the decomposition by sources of the Zenga 1984 point and synthetic inequality indexes

In this work we provide a decomposition by sources of the inequality index \(\zeta \) defined by Zenga (Giornale degli Economisti e Annali di economia 43(5–6):301–326, 1984). The source contributions are obtained with the method proposed in Zenga et al. (Stat Appl X(1):3–31, 2012) and Zenga (Stat Appl XI(2):133–161, 2013), that allows to compare different inequality measures. This method is based on the decomposition of inequality curves. To apply this decomposition to the index \(\zeta \) and its inequality curve, we adapt the method to the “cograduation” table. Moreover, we consider the case of linear transformation of sources and analyse the corresponding results.     

A new generalized Weibull distribution in income economic inequality curves

Researchers have been developing various extensions and modified forms of the Weibull distribution to enhance its capability for modeling and fitting different data sets. In this note, we investigate the potential usefulness of the new modification to the standard Weibull distribution called odd Weibull distribution in income economic inequality studies. Some mathematical and statistical properties of this model are proposed. We obtain explicit expressions for the first incomplete moment, quantile function, Lorenz and Zenga curves and related inequality indices. In addition to the well-known stochastic order based on Lorenz curve, the stochastic order based on Zenga curve is considered. Since the new generalized Weibull distribution seems to be suitable to model wealth, financial, actuarial and especially income distributions, these findings are fundamental in the understanding of how parameter values are related to inequality. Also, the estimation of parameters by maximum likelihood and moment methods is discussed. Finally, this distribution has been fitted to United States and Austrian income data sets and has been found to fit remarkably well in compare with the other widely used income models.     

On the parameters of Zenga distribution

In 2010 Zenga introduced a new three-parameter model for distributions by size that can be used to represent income, wealth, financial and actuarial variables. This paper proposes a summary of its main properties, followed by a focus on the interpretation of the parameters in terms of inequality. The scale parameter μ is equal to the expectation, and it does not affect the inequality, while the two shape parameters α and θ are inverse and direct inequality indicators respectively. This result is obtained through stochastic orders based on inequality curves. A procedure to generate a random sample from Zenga distribution is also proposed. The second part of this article looks at the parameter estimation. Analytical solution of method of moments is obtained. This result is used as a starting point of numerical procedures to obtain maximum likelihood estimates both on ungrouped and grouped data. In the application, three empirical income distributions are considered and the aforementioned estimates are evaluated. A comparison with other well-known models is provided, by the evaluation of three goodness-of-fit indexes.     

The New Zenga curve in the context of reliability analysis

ABSTRACT

In this article, we look into the properties and characterizations of the New Zenga curve. The relationship of the curve with other measures of inequality as well as some reliability concepts are examined. Classification of lifetime distributions using the Zenga curve and an illustration for the behaviour of the curve using a survival data are also provided.     

Comparing point and interval estimates in the bivariate <Emphasis Type="Italic">t</Emphasis>-copula model with application to financial data

The paper considers joint maximum likelihood (ML) and semiparametric (SP) estimation of copula parameters in a bivariate t-copula. Analytical expressions for the asymptotic covariance matrix involving integrals over special functions are derived, which can be evaluated numerically. These direct evaluations of the Fisher information matrix are compared to Hessian evaluations based on numerical differentiation in a simulation study showing a satisfactory performance of the computationally less demanding Hessian evaluations. Individual asymptotic confidence intervals for the t-copula parameters and the corresponding tail dependence coefficient are derived. For two financial datasets these confidence intervals are calculated using both direct evaluation of the Fisher information and numerical evaluation of the Hessian matrix. These confidence intervals are compared to parametric and nonparametric BCA bootstrap intervals based on ML and SP estimation, respectively, showing a preference for asymptotic confidence intervals based on numerical Hessian evaluations.     

Multiple comparisons with a control for exponential location parameters under heteroscedasticity

In this paper, a new design-oriented two-stage two-sided simultaneous confidence intervals, for comparing several exponential populations with control population in terms of location parameters under heteroscedasticity, are proposed. If there is a prior information that the location parameter of k exponential populations are not less than the location parameter of control population, one-sided simultaneous confidence intervals provide more inferential sensitivity than two-sided simultaneous confidence intervals. But the two-sided simultaneous confidence intervals have advantages over the one-sided simultaneous confidence intervals as they provide both lower and upper bounds for the parameters of interest. The proposed design-oriented two-stage two-sided simultaneous confidence intervals provide the benefits of both the two-stage one-sided and two-sided simultaneous confidence intervals. When the additional sample at the second stage may not be available due to the experimental budget shortage or other factors in an experiment, one-stage two-sided confidence intervals are proposed, which combine the advantages of one-stage one-sided and two-sided simultaneous confidence intervals. The critical constants are obtained using the techniques given in Lam [9,10]. These critical constant are compared with the critical constants obtained by Bonferroni inequality techniques and found that critical constant obtained by Lam [9,10] are less conservative than critical constants computed from the Bonferroni inequality technique. Implementation of the proposed simultaneous confidence intervals is demonstrated by a numerical example.     

A note on simultaneous inference with rerandomization tests

Valid simultaneous confidence intervals based on rerandomization are provided for the first time. They are derived from joint confidence regions which are constructed by testing for all possible parametric values. A simple exampe illustrates these confidence intervals and compares inferences from them with other methods.     

Improved simultaneous confidence intervals for linear contrasts and regression parameters

A well known method for obtaining conservative simultaneous confidence intervals for the K parameters in a linear regression model, or for K linear contrasts, is based on the percentage points of the Studentized maximum modulus distribution. From an inequality due to Sidak, conservative yet uniformly shorter confidence intervals would be possible if the percentage points of a particular form of the multivariate t distribution were available. The purpose of this paper is to provide the required percentage points. For K<8 the resulting confidence intervals can be substantially shorter.     

Asymptotic results in partially non-regular log-exponential distributions

We propose approximations to the moments, different possibilities for the limiting distributions and approximate confidence intervals for the maximum-likelihood estimator of a given parametric function when sampling from partially non-regular log-exponential models. Our results are applicable to the two-parameter exponential, power-function and Pareto distribution. Asymptotic confidence intervals for quartiles in several Pareto models have been simulated. These are compared to asymptotic intervals based on sample quartiles. Our intervals are superior since we get shorter intervals with similar coverage probability. This superiority is even assessed probabilistically. Applications to real data are included.     

Empirical likelihood confidence intervals for the mean of a population containing many zero values

If a population contains many zero values and the sample size is not very large, the traditional normal approximation‐based confidence intervals for the population mean may have poor coverage probabilities. This problem is substantially reduced by constructing parametric likelihood ratio intervals when an appropriate mixture model can be found. In the context of survey sampling, however, there is a general preference for making minimal assumptions about the population under study. The authors have therefore investigated the coverage properties of nonparametric empirical likelihood confidence intervals for the population mean. They show that under a variety of hypothetical populations, these intervals often outperformed parametric likelihood intervals by having more balanced coverage rates and larger lower bounds. The authors illustrate their methodology using data from the Canadian Labour Force Survey for the year 2000.     

Some remarks on Lorenz ordering-preserving functionals

Basing on two well-known characterization results on stochastic dominance and continuous majorization relation, the ordering-preserving property-with respect to Lorenz ordering-is deduced for a wide class of families of functionals on a class of distributions. As a consequence the isotonicity ofZ Zenga concentration index is deduced as an immediate application of a characterization result, in particular of the first degree stochastic dominance relation. Moreover it is also shown that a classical inequality by Fan and Lorenz is a basic reference for the determination of a wide class of Lorenz ordering-preserving functionals. Isotonicity ofZ could also be seen as a straighforward application of Fan and Lorenz inequality.     

Simultaneous confidence intervals for comparing several inverse Gaussian means under heteroscedasticity

Recently, Zhang [Simultaneous confidence intervals for several inverse Gaussian populations. Stat Probab Lett. 2014;92:125–131] proposed simultaneous pairwise confidence intervals (SPCIs) based on the fiducial generalized pivotal quantity concept to make inferences about the inverse Gaussian means under heteroscedasticity. In this paper, we propose three new methods for constructing SPCIs to make inferences on the means of several inverse Gaussian distributions when scale parameters and sample sizes are unequal. One of the methods results in a set of classic SPCIs (in the sense that it is not simulation-based inference) and the two others are based on a parametric bootstrap approach. The advantages of our proposed methods over Zhang’s (2014) method are: (i) the simulation results show that the coverage probability of the proposed parametric bootstrap approaches is fairly close to the nominal confidence coefficient while the coverage probability of Zhang’s method is smaller than the nominal confidence coefficient when the number of groups and the variance of groups are large and (ii) the proposed set of classic SPCIs is conservative in contrast to Zhang’s method.     

含零总体中位数的经验似然

当所研究的总体中含有一定比率的零值,而其余的值大于零时,则称为含零总体。文章利用Owen的经验似然方法来构造这类总体的中位数的置信区间,所得结果不需要假设总体服从某种参数模型,并能利用样本中零值的信息,而且构造的置信区间受非零总体偏斜的影响比一些其他非参数方法小一些。同时,随机模拟的结果也显示了这一点。     

Simultaneous confidence intervals in the analysis of orthogonal saturated designs

We present the first known method of constructing exact simultaneous confidence intervals for the analysis of orthogonal, saturated factorial designs. Given m independent, normally distributed, unbiased estimators of treatment contrasts, if there is an independent chi-squared estimator of error variance, then simultaneous confidence intervals based on the Studentized maximum modulus distribution are exact under all parameter configurations. In this paper, an analogous method is developed for the case of an orthogonal saturated design, for which the treatment contrasts are independently estimable but there is no independent estimator of error variance. Lacking an independent estimator of the error variance, the smallest sums of squares of effect estimators are pooled. The simultaneous confidence intervals are based on a probability inequality, for which the simultaneous confidence coefficient is achieved in the null case.     

Empirical Likelihood Intervals for Conditional Value‐at‐Risk in Heteroscedastic Regression Models

Abstract. Non‐parametric regression models have been studied well including estimating the conditional mean function, the conditional variance function and the distribution function of errors. In addition, empirical likelihood methods have been proposed to construct confidence intervals for the conditional mean and variance. Motivated by applications in risk management, we propose an empirical likelihood method for constructing a confidence interval for the pth conditional value‐at‐risk based on the non‐parametric regression model. A simulation study shows the advantages of the proposed method.     

EMPIRICAL LIKELIHOOD METHODS FOR THE GINI INDEX

The Gini index and its generalizations have been used extensively for measuring inequality and poverty in the social sciences. Recently, interval estimation based on nonparametric statistics has been proposed in the literature, for example the naive bootstrap method, the iterated bootstrap method and the bootstrap method via a pivotal statistic. In this paper, we propose empirical likelihood methods to construct confidence intervals for the Gini index or the difference of two Gini indices. Simulation studies show that the proposed empirical likelihood method performs slightly worse than the bootstrap method based on a pivotal statistic in terms of coverage accuracy, but it requires less computation. However, the bootstrap calibration of the empirical likelihood method performs better than the bootstrap method based on a pivotal statistic.     

Asymptotic Confidence Intervals for a New Inequality Measure

This work aims at assessing, by simulation methods, the performance of asymptotic confidence intervals for Zenga's new inequality measure. The results are compared with those obtained on Gini's measure, perhaps the most widely used index for measuring inequality in income and wealth distributions. Our findings show that the coverage accuracy and the size of the confidence intervals for the two measures are very similar in samples from economic size distributions.     

Inferences on the common mean of several heterogeneous log-normal distributions

We consider the problem of making inferences on the common mean of several heterogeneous log-normal populations. We apply the parametric bootstrap (PB) approach and the method of variance estimate recovery (MOVER) to construct confidence intervals for the log-normal common mean. We then compare the performances of the proposed confidence intervals with the existing confidence intervals via an extensive simulation study. Simulation results show that our proposed MOVER and PB confidence intervals can be recommended generally for different sample sizes and number of populations.